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For (most) PIDs: Trace zero matrices are commutators

Let $R$ be a commutative ring and $M_n(R)$ denote the ring of $n\times n$ matrices with coefficients in $R$. For $X,Y\in M_n(R)$, their commutator $[X,Y]$ is defined by $$[X,Y] := XY – YX.$$ The trace of any matrix is defined as the sum of its diagonal entries. If $X$ and $Y$ are any matrices, what […]